### Recall from last lecture:

## The Yukawa Theory of Quantum Exchange

This section and the next present a "hand waving" argument for the form of the propagator expression.
The following is my summary of the argument.

The binding of protons and neutrons in a nucleus was postulated to be due to a short range strong force with a potential of the form:

U(r) = (g_{0}/4pr)e^{-r/R}

where R can be related to a mass by
R = hbar/mc.

Without the exponential term, the form of the potential is like the electrostatic potential U = Q/4pr, leading us to interpret g_{0} as a sort of "strong force charge".
Now, we simply call g_{0} the strong coupling constant.
The mass in R is interpreted as the mass of the boson exchanged between two particles, and, for m>0, it makes the force short range.
Yukawa's theory now is seen as an approximation to the true effects of the strong force with massless gluons.
## The Boson Propagator

This idea resurfaces in modern quantum field theory.
Consider the interaction of a particle with a heavy source of force.
The electron and nucleus exchange a boson, altering the electron's momentum by an amount **q**.
The probability for momentum transfer **q** is just the Fourier transform, f(**q**), of the potential U(**r**).
(Demonstration of this is covered in the field theory course.)

f(**q**) = g *Re*{Int[U(**r**) e^{iq·r} dV]}

where the electron couples to the boson with strength g.
For a central potential, U(**r**) = U(r), and (after a few steps) the integration yields
f(**q**) = gg_{0} /(|**q**|^{2} + m^{2})

This is the momentum space description of the interaction with a heavy source that couples to the boson with strength g_{0}.
We used a heavy source so that we didn't have to worry about the recoil of the source, and it made the integral non-relativistic.
In the general case, we need an expression with a covariant form, and we get that by changing |**q**|^{2} to the four vector squared:

f(q^{2}) = g g_{0} / (m^{2} - q^{2})

(Note that the difference in sign with Eq. 2.6 in Perkins is due to the different relativistic metric.)
This expression says that the probability for two particles to exchange a boson with 4-momentum transfer q^{2} is given by the product of the boson couplings to the particles divided by the difference between the mass squared of the (real) boson and the q^{2}.
When the q^{2} equals m^{2}, the probability becomes large.
For photons and gluons, this occurs for q^{2}=0.
For W^{±} and Z°, this occurs around q^{2}=90GeV!

Copyright © Robert Harr 2003